intersection of 2 vectors in homogenous space

Question

This is a homework question but i would just like to clarify my solution

Given a vector

$$L_1: (1, 1, 2) + x(1, 4, 1)$$ $$L_2: (2, 2, 3) + y(1, 4, 1)$$

Finding the intersection of these $2$ lines in projective geometry will result in cross product of

$$L_1 \times L_2$$

Hence, the solution would be

$$(1, 0, 0) + z(0, 0, 0)$$

Am I right to say this?

Thanks.